434 
EUCLID 
the two senses of the word rropLaya. The first is that of 
a corollary, where something appears as an incidental result 
of a proposition, obtained without trouble or special seeking, 
a sort of bonus which the investigation has presented us 
with. 1 The other sense is that of Euclid’s Porisms. In 
this sense 
‘ porism is the name given to things which are sought, but 
need some finding and are neither pure bringing into existence 
nor simple theoretic argument. For (to prove) that the angles 
at the base of isosceles triangles are equal is matter of theoretic 
argument, and it is with reference to things existing that sucli 
knowledge is (obtained). But to bisect an angle, to construct 
a triangle, to cut off, or to place—all these things demand the 
making of something; and to find the centre of a given circle, 
or to find the greatest common measure of two given commen 
surable magnitudes, or the like, is in some sort intermediate 
between theorems and problems. For in these cases there is 
no bringing into existence of the things sought, but finding 
of them; nor is the procedure purely theoretic. For it is 
necessary to bring what is sought into view and exhibit it 
to the eye. Such are the porisms which Euclid wrote and 
arranged in three books of Porisms.’ 2 
Proclus’s definition thus agrees well enough with the first, 
the ‘ older ’, definition of Pappus. A porism occupies a place 
between a theorem and a problem; it deals with something 
already existing, as a theorem does, but has to find it (e.g. the 
centre of a circle), and, as a certain operation is therefore 
necessary, it partakes to that extent of the nature of a problem, 
which requires us to construct or produce something not 
previously existing. Thus, besides III. 1 and X, 3, 4 of the 
Elements mentioned by Proclus, the following propositions are 
real porisms: III. 25, VI. 11-13, YU. 33, 34, 36, 39, VIII. 2, 4, 
X. 10, XIII. 18. Similarly, in Archimedes's On the sphere and 
Cylinder, I. 2-6 might be called porisms. 
The enunciation given by Pappus as comprehending ten of 
Euclid’s propositions may not reproduce the form of Euclid’s 
enunciations; but, comparing the result to be proved, that 
certain points lie on straight lines given in position, with the 
class indicated by II above, where the question is of such and 
such a point lying on a straight line given in position, and 
1 Proclus on Each I, pp. 212. 14; 301. 22. 2 lb., p. 301. 25 sq.